Langmuir 2006, 22, 6161-6167
6161
Directing Droplets Using Microstructured Surfaces†
Ashutosh Shastry,* Marianne J. Case, and Karl F. Böhringer
Department of Electrical Engineering, UniVersity of Washington, Seattle, Washington 98195
ReceiVed January 17, 2006. In Final Form: March 8, 2006
Systematic variation of microscale structures has been employed to create a rough superhydrophobic surface with
a contact angle gradient. Droplets are propelled down these gradients, overcoming contact angle hysteresis using
energy supplied by mechanical vibration. The rough hydrophobic surfaces have been designed to maintain air traps
beneath the droplet by stabilizing its Fakir state. Dimensions and spacing of the microfabricated pillars in silicon
control the solid-liquid contact area and are varied to create a gradient in the apparent contact angle. This work
introduces the solid-liquid contact area fraction as a new control variable in any scheme of manipulating droplets,
presenting theory, fabricated structures, and experimental results that validate the approach.
1. Introduction
Microscale bioanalysis is pregnant with assay possibilities
that are potentially useful in diagnostic applications and as research
tools for biologists. Continuous flow systems have been the default
approach toward such lab-on-chip bioassay systems. Dropletbased lab-on-chip applications have become increasingly popular,
however, because they enable spatially and temporally resolved
chemistries.2 Employing surface-energy gradients to cause the
movement of droplets has been the common theme of several
apparently disparate actuation approaches.3 These surface-energy
gradients can be categorized into static or hard-coded and dynamic
or “programmable”. Electrowetting is an example of a programmable surface-energy gradient.4,5 There are other instances where
the gradient tracks are hard-coded but the actuation is programmable,6,7 employing the Marangoni effect for droplet actuation.
The droplet is confined to a hard-coded hydrophilic track and
propelled along by a dynamic temperature gradient.
Chaudhary and Whitesides8 created a passive contact angle
gradient by reacting silicon with the diffusing front of alkyltrichlorosilane, generating a gradually varying chemical composition along the length of the surface. The highlight of the
work was the very low hysteresis surface that ensured that the
force of the gradient alone was sufficient to move the droplet
against gravity. Using similar chemical gradients, Daniel et al.
actuated the droplets harnessing alternative sources of energy to
overcome the impeding force of contact angle hysteresis. In one
case, the energy released by the condensation of steam propelled
the drops along radial energy gradients,9 whereas acoustic
vibration was employed in other cases.10,11 Bain et al.12 introduced
dynamic gradients formed by the continuous chemical reaction
† Preliminary results from section 5.1 were presented at the IEEE
International Conference on MEMS in Jan 2005.1
* Corresponding author. E-mail: ashutosh@washington.edu. Phone:
(206) 543 5218.
(1) Shastry, A.; Case, M. J.; Böhringer, K. F. Engineering Surface Roughness
to Manipulate Droplets in Microfluidic Systems. 18th IEEE International Conference
on Micro Electro Mechanical Systems, Miami, FL, 2005.
(2) Stone, H. A.; Stroock, A. D.; Ajdari, A. Annu. ReV. Fluid Mech. 2004, 36,
381-411.
(3) Darhuber, A. A.; Troian, S. M. Annu. ReV. Fluid Mech. 2005, 37, 425455.
(4) Mugele, F.; Baret, J. C. J. Phys.: Condens. Matter 2005, 17, R705-R774.
(5) Pollack, M.; Fair, R.; Shenderov, A. Lab Chip 2002, 2, 96-101.
(6) Kataoka, D. E.; Troian, S. M. Nature 1999, 402, 794-797.
(7) Brochard, F. Langmuir 1989, 5, 432-438.
(8) Chaudhary, M. K.; Whitesides, G. M. Science 1992, 256, 1539-1541.
(9) Daniel, S.; Chaudhury, M. K. Chen, J. C. Science 2001, 291, 633-636.
(10) Daniel, S.; Chaudhury, M. K. Langmuir 2002, 18, 3404-3407.
of hydrophobic alkyl silanes inside the droplet with the solid
surface. Thiele et al.13 further explored such dynamic gradients
through modeling and experiments.
In recent years, there have been remarkable advances in our
understanding of the wetting of textured surfaces.14,15 Mahadevan
introduced the term “Fakir”,16 and later Quéré used it to describe
droplets resting on rough surfaces with air pockets trapped beneath
them.17 These trapped air pockets ensure a low solid-liquid
contact area fraction and significantly reduced drag,19,20 allowing
droplets to be moved on the surface with low amounts of energy.21
Fakir droplets are very stable on superhydrophobic nanorough
surfacessthey are able to resist collapsing under large pressures.22
Their stability and low resistance to flow make Fakir droplets
attractive candidates for any low-energy droplet manipulation
scheme. Petrie et al.23 recently reported the first efforts to
manipulate these Fakir droplets, employing Chaudhary’s static
chemical gradient approach on a nanorough surface that stabilizes
Fakir droplets. Lee et al.24 have recently reported a device that
switches the roughness of a membrane to move a droplet.
We demonstrate, for the first time, droplets moving down
surface-energy gradients created by systematically varying the
solid-liquid area fraction of Fakir droplets.1 We start with a
short review of the theory of wetting of rough surfaces and droplet
movement due to contact angle gradients. We combine the two
theories to propose a model that predicts conditions for the
incipient motion of droplets on roughness-controlled gradient
tracks. Next, we outline our design approach, fabrication process,
(11) Daniel, S.; Sircar, S.; Gliem, J.; Chaudhury, M. K. Langmuir 2004, 20,
4085-4092.
(12) Bain, C. D.; Burnett-Hall, G. D.; Montgomerie, R. R. Nature 1994, 372,
415-416.
(13) Thiele, U.; John, K.; Bär, M. Phys. ReV. Lett. 2004, 93, 027802-1027802-4.
(14) Bico, J.; Marzolin, C.; Quéré, D. Europhys. Lett. 1999, 47, 220-226.
(15) Bico, J.; Thiele, U.; Quéré, D. Colloids Surf., A 2002, 206, 41-46.
(16) Mahadevan, L. Nature 2001, 411, 895-896.
(17) Quéré, D. Nat. Mater. 2002, 14, 1109-1112.
(18) Lafuma, A.; Quéré, D. Nat. Mater. 2003, 2, 457-460.
(19) Cottin-Bizonne, C.; Barrat, J. L.; Bocquet, L.; Charlaix, E. Nat. Mater.
2003, 2, 237-240.
(20) Choi, C. H.; Kim, C. J. Measurement of Slip on Nanotur Surfaces.
Proceedings of ASME 2005: Integrated Nanosystems Design, Synthesis &
Applications, Berkeley, CA, 2005.
(21) Quéré, D.; Lafuma, A.; Bico, J. Nanotechnology 2003, 1, 14-15.
(22) Journet, C.; Moulinet, S.; Ybert, C.; Purcell, S. T.; Bocquet, L. Europhys.
Lett. 2005, 71, 104-109.
(23) Petrie, R. J.; Bailey, T.; Gorman, C. B.; Genzer, J. Langmuir 2004, 20,
9893-9896.
(24) Lee, J.; He, B.; Patankar, N. A. J. Micromech. Microeng. 2005, 15, 591600.
10.1021/la0601657 CCC: $33.50 © 2006 American Chemical Society
Published on Web 06/09/2006
6162 Langmuir, Vol. 22, No. 14, 2006
Shastry et al.
Figure 1. Texture parameters φ and r expressed in terms of design
parameters a (gap length), b (pillar size), and h (pillar height), where
φ is the fraction of the pillar top area over the total horizontal surface
area and r is the fraction of the total surface area over the total
horizontal surface area. The value φ determines the apparent contact
angle θF as given by eq 3. The value r correlates with the relative
stability of a droplet in the Fakir state as opposed to the Wenzel
state.
Figure 3. Schematic showing energy levels of the Fakir and Wenzel
states. The choice of texture parameters makes the Fakir state
metastable in this example. The energy of the intermediate state is
calculated by assuming nearly complete penetration of the droplet;
only a thin film of air separates the liquid-air and solid-air interfaces
at the bottoms of the valleys. A Fakir droplet needs to overcome the
energy barrier Ebarrier,F ) EIS - EF to move to the Wenzel state.
Fakir state, the base of the droplet essentially contacts a composite
surface of pillar tops and air. The apparent contact angle θCB on
a composite surface is determined using the Cassie-Baxter
relation given in eq 2
cos θCB )
Figure 2. Droplets of volume 5 µL on a Teflon-coated silicon
surface with φ ) 0.05 and r ) 1.4 in (a) the Wenzel state with a
footprint diameter of 1.96 mm, θW ) 118° (expected, 112.8°) and
(b) the Fakir state with a footprint diameter of 1 mm, θF ) 156.6°
(expected, 164.5°). Air pockets are visible between pillars under the
Fakir-state droplet. In our experiments as shown in these figures,
pinning of the droplet edge caused significant deviations from the
predicted equilibrium value.
and experimental setup. Finally, we present results and observed
trends and compare them with the theoretical predictions based
on our model. We conclude by pointing out the implications of
this work and resulting directions that emerge as worthwhile
pursuits.
2. Theory and Model
We begin by reviewing the theory of wetting of rough surfaces.
Then we develop a model of the forces acting on the droplet
resting on a roughness-controlled contact angle gradient. Next,
we use this model to estimate the minimum slope required for
the droplet to move without the need for external forces.
2.1. Review of the Wetting of Rough Surfaces. Consider a
rough surface realized by creating pillars of controlled geometry
on a smooth surface. Here the roughness is characterized by r,
the ratio of rough to planar surface area, as explained in Figure
1. The basic effect of surface roughness on wetting is modeled
by Wenzel’s relation (eq 1), which relates the apparent contact
angle θW of a droplet on a rough surface with r g 1 to Young’s
intrinsic contact angle θi.25
cos θW ) r cos θi
(1)
In the Wenzel state, the droplet is conformal with the
topography, as seen in Figure 2a. The droplet can also sit on the
pillar tops with air pockets trapped beneath it, as shown in Figure
2b. This configuration is referred to as the “Fakir” state. In the
(25) de Gennes, P. G.; Brochard-Wyart, F.; Quéré, D. Capillarity and Wetting
Phenomena: Drops, Bubbles, Pearls, WaVes; Springer: New York, 2004.
∑φj cos θj
(2)
where φj is the surface area fraction and θj is the intrinsic contact
angle of material j with water.
For a sessile Fakir droplet, a surface area fraction φ (Figure
1) of its base rests on pillar tops that have an intrinsic contact
angle of θi; the remaining surface area fraction of (1 - φ) is
freely suspended and is in contact with air with a contact angle
of 180°. Substituting these values, the apparent contact angle in
the Fakir state, a special case of Cassie-Baxter contact, is readily
computed14,15 to yield eq 3.
cos θF ) φ(cos θi + 1) - 1
(3)
Angles θF from eq 3 and θW from eq 1 are equilibrium values
of the apparent contact angle in the two states. Equilibrium angles
of the droplet are expected when there is no impending movement.
Even under these conditions, there are significant fluctuations
of the measured contact angle, also called contact angle
hysteresis,26 when the two energy levels are close. Contactangle hysteresis is defined as the difference between the cosines
of maximum advancing and minimum receding angles that a
droplet makes with a surface. Hysteresis results from the pinning
of the three-phase contact line to the solid surface and is attributed
to physical and chemical inhomogeneities.25 We also notice by
comparing θF from eq 3 and θW from eq 1 that the Fakir state
has a lower energy relative to the Wenzel state (i.e., cos θW <
cos θF) if the following inequality proposed by Bico et al.15 (eq
4) holds true.
cos θi <
φ-1
r-φ
(4)
A droplet at a given location on a surface does not spontaneously transit from one state to the other because of the presence
of an energy barrier. This energy barrier is analogous to the
activation energy of a chemical reaction that prevents spontaneous
conversion to products (Figure 3). This energy barrier is easily
(26) Lafuma, A.; Quéré, D. Nat. Mater. 2003, 2, 457-460.
Directing Droplets Using Microstructured Surfaces
Langmuir, Vol. 22, No. 14, 2006 6163
of liquid of width dy experiences a driving force parallel to x,
following Brochard.7 The slice occupies an interval xb < x < xf.
The following equation (eq 6) represents the force balance under
the condition of incipient movement of the slice:
dUnet,slice ) γLV(cos θb,r - cos θf,a) dxdy ) 0
(6)
Since the contact angles of the two edges are equal, there is
no pressure gradient and hence no flow at the threshold of
movement. The threshold condition represents an equilibrium
between the driving force and hysteresis force such that the net
force is zero:
Figure 4. Droplet slice of width dy shown moving a length dx down
a contact angle gradient: (a) top view and (b) side view. The previous
location of the droplet is shown in light gray. A film is left behind
on previous pillars as shown, and the front area of the droplet increases
by SLV, which is represented by a teal patch.
estimated.27 In situations where eq 4 does not hold, the energy
barrier gives a useful bound on the energy that needs to be coupled
to a Fakir droplet before risking its transition to the Wenzel state.
2.2. Model for the “Slope” Required for Incipient Movement. Let us examine a sessile droplet at rest on a horizontal
hydrophobic surface. The total energy U of a droplet is the sum
of two components: Uγ due to its interfaces and Ug due to its
position in a force field of body forces acting on it. We consider
a small water droplet of radius R < κ-1 where F is the density,
g is the acceleration due to gravity, γLV is the surface tension
of the liquid, and κ-1 is the capillary length xγLV/Fg. For water,
κ-1 is 2.7 mm, corresponding to approximately 80 µL of water
in a spherical drop. For droplets smaller than 80 µL, gravity has
a negligible effect on shape, and the droplet can be approximated
as a spherical cap. In the absence of any body forces, Ug ≡ 0
and the following equation (eq 5) is true.
U ) Uγ ) ASLγSL + ASVγSV + ALVγLV
(5)
When the droplet rests on a surface energy gradient, it
experiences a driving force. Spontaneous movement down the
gradient is impeded, however, by the opposing force due to contact
angle hysteresis. When an external force is applied to mitigate
hysteresis, the droplet moves down the surface energy gradient,
overcoming pinning. When the droplet is in motion, the viscous
drag adds to the hysteresis force to oppose the driving force. But
under the condition of incipient motion, which occurs just before
the droplet moves, viscous drag is nonexistent, and the driving
force equals the hysteresis force.
Let us consider the droplet shown in Figure 4. The height of
the droplet slice is a function of the x coordinate only. A slice
(27) Patankar, N. A. Langmuir 2004, 20, 7097-7102.
To model the incipient motion of the droplets, we need to
express each term in eq 7 as a function of surface parameters and
integrate it over the footprint of the droplet. In the absence of
rigorous theoretical models for θr and θa on rough hydrophobic
surfaces, we employ the heuristic relations available from the
literature to predict the bounds of advancing and receding contact
angles after having experimentally validated them.30
To obtain θr, we follow the approach of Fort and Roura28,29
who postulate that a thin film of liquid is left behind as shown
in Figure 4. This assumption, originally proposed for hydrophilic
surfaces,28 expectedly overestimates hysteresis, providing an
upper bound for hydrophobic surfaces, except for the low φ
regime.
To obtain θa, we consider a droplet moving forward while its
three phase contact line remains pinned, resulting in an advancing
angle instead of the equilibrium angle at the front edge. The
deviation from the equilibrium value is captured through a factor
SLV which equals cos θ - cos θa and accounts for the increased
liquid-vapor interface shown in Figure 4 at the front edge. It
is also the measure of energy required for depinning. Once the
maximum advancing angle is reached, the edge gets depinned
and advances.
To estimate the driving force, we displace the whole slice by
dx and write an expression for the variation of interfacial energy
dU. The expression for the differential energy dU of the slice
follows in eq 8:
φf and φb are values of φ at the front and back edges of the
droplet, respectively, and the interfacial energy symbols γij
maintain their previously defined meanings.
(28) Roura, P.; Fort, J. Phys. ReV. E 2001, 64, 011601.
(29) Roura, P.; Fort, J. Langmuir 2002, 18, 566-569.
(30) Shastry, A.; Goyal, S.; Epilepsia, A.; Case, M. J.; Abbasi, S.; Ratner, B.
D.; Böhringer, K. F. Engineering Surface Micro-structure to Control Fouling and
Hysteresis in Droplet based Microfluidic Bioanalytical Systems. In the Technical
Digest of the 12th Solid State Sensor, Actuator and Microsystems Workshop,
Hilton Head, June 4-8, 2006.
6164 Langmuir, Vol. 22, No. 14, 2006
Shastry et al.
Next we use Young’s equation (cos θi ) (γSV - γSL)/γLV) and
rearrange eq 8 to obtain the following expression (eq 9) for the
net force on the slice, Fnet,slice.
The first term (2φb - 1) equals cos θb,r, where cos θb,r is the
receding angle of the back edge of the droplet as obtained by
Patankar.31 As mentioned earlier, recent experimental study by
Shastry et al.30 shows that this model provides an upper bound
for φ > 0.1.
In the absence of models to predict SLV, we substitute it with
cos θF - cos θF,a and employ the heuristic relation proposed by
He et al.32 given in eq 10 for the advancing contact angle θF,a
of a Fakir droplet
cos θF,a ) -1 + φF(cos θi,a + 1)
(10)
where θi,a is the advancing contact angle on a smooth surface.
We then write Fnet,slice, in terms of φ and the intrinsic contact
angle θi by starting with eq 7, using the contact angle for the
Fakir droplet (eq 3), and substituting for cos θf,a from eq 10 and
cos θb,r from eq 9 to obtain eq 11.
Figure 5. Teflon-coated continuous gradient with θi ) 115°. The
values of intrinsic advancing and receding angles used in the hysteresis
model were the measured values θi,r ) 113° and θi,a ) 160°. (a) The
slope required for incipient motion given φc at location xc on the
gradient and the actual slope of the gradient at each position xc are
plotted for a 5 µL droplet. (b) For the same gradient and droplet
volume, the magnitudes of the driving force Fd(x) and hysteresis
force Fhyst(x) as obtained from eq 16 are shown.
Now substituting for Fnet,slice from eq 11, we obtain the
following (eq 15)
At this stage, we have expressions for the driving force and
hysteresis force on the slice as functions of φf and φb. We need
to express φf and φb as functions of the position of the droplet
on the gradient. We do so by utilizing the constant slope of the
gradient tracks an approach parallel to those of Brochard7 and
Daniel et al.10 Let φc be the value of φ at xc, the center of the
slice that occupies an interval xb < x < xf. For the slice, xb )
xc - ζ and xf ) xc + ζ, hence -(dφ/dx)2ζ ) (φb - φf). The
following substitutions can therefore be made.
φb ) φc(x) -
dφ
ζ
dx
(12)
φf ) φc(x) +
dφ
ζ
dx
(13)
where C ) (dφ/dx)(2 cos θi - 1 - cos θi,a)
The final force expression is thus obtained (eq 16).
It is important to notice that R depends on both droplet volume
V and location xc depends on the gradient. R is approximated by
the following equation (eq 17).
R ) sin θc
We have thus expressed forces on a slice in terms of its location
on the gradient. To estimate forces on the entire droplet, we
integrate these forces over the range of y, which is 2R, the diameter
of the footprint of the droplet, noting that the slope dφ/dx is a
constant.
Fnet,droplet ) 2
RFnet,slice
∫0
dy
dy
(14)
(31) Patankar, N. A. Langmuir 2003, 19, 1249-1253.
(32) He, B.; Lee, J.; Patankar, N. A. Colloids Surf., A 2004, 248, 101-104.
(
)
3V
π(cos θc - 3cos θc + 2)
3
1/3
(17)
A plot of these force contributions as a function of position
for a typical gradient is shown in Figure 5b.
3. Design and Fabrication
3.1. Design. We have developed chemically homogeneous
but textured surfaces. A regular two-dimensional array of square
pillars created a rough surface. The dimensions of the surface,
with gap length a and pillar width b, were varied spatially across
distances ranging from 4 to 8 mm to produce several texture
gradients with uniform linear changes in cos θF along their length
Directing Droplets Using Microstructured Surfaces
Langmuir, Vol. 22, No. 14, 2006 6165
N-type ⟨100⟩ silicon wafers as the substrate. Photolithography
was performed using an AZ4620 photoresist. Next, pillars were
etched using the standard Bosch process for deep reactive ion
etching (DRIE). The etch depth defined pillars of height 40 µm.
The SEM micrograph in Figure 7c shows etched pillars after
stripping the resist slice. The rough surfaces were then coated
with an adhesion-promoting silane and finally with Teflon
AF1600 to obtain the hydrophobic layer.
4. Setup and Experiment
The experimental setup is shown in Figure 8. A gradient die was
secured to a stage using double-sided tape. The stage was glued to
the diaphragm of a speaker, which was in turn connected to a function
generator (Agilent 33120A). A square wave (typically 20 Hz, 50
mV) was applied to the speaker using the function generator, causing
the stage to vibrate vertically with a typical amplitude of 180 µm
as shown in Figure 8. Water droplets ranging from 2 to 10 µL were
deposited at the beginning of the gradient using a hand syringe. The
droplet was illuminated with a 300 W halogen lamp (Gemini highintensity light source) directed through an optical fiber bundle. There
was no observable evaporation of the droplet during the time it took
the droplet to move down the gradient (<1-8 s). A high-speed
camera (Lightning RDT, Data and Imaging Systems) recorded the
movement at 500 frames/s.
5. Results and Discussion
Figure 6. (a) Contact angle gradient having an approximately
constant slope (d cos θF)/dx. The minimum and maximum contact
angles were determined by the limits of the design space, using θi,a
) 115° for Teflon. Pillar widths and gaps were specified to create
a gradient of specified length between the two end points. (b) Contour
plot showing the possible gap length and pillar width combinations
within the constraints imposed by design and fabrication. Vertical
isolines represent combinations of constant θF. Isocolors specify
combinations with constant Ebarrier,F. The bold orange line signifies
the combinations for which Ebarrier,W ) Ebarrier,F, as given by eq 4.
It is desirable to maintain Ebarrier,F > Ebarrier,W to ensure that Fakir
droplets remain stable.
(Figure 6a). In some cases, a steeper slope was used on the latter
section of the gradient to mitigate the increasing hysteresis force
associated with larger φ values.
Fabrication constraints and design requirements decided the
limits of φ, φmin, and φmax. The constraints on the design space
are explained in Figure 6b. The desired cos θF at each location
along the gradient was a function of the intended gradient length
and slope. Actual gradient length and slope varied slightly
depending on the (a, b) combinations employed. As seen in
Figure 6b, each desired contact angle can be realized using several
(a, b) combinations. Because the resolution of the photolithography mask was limited to 1 µm, the most precise values of (a,
b) are those close to whole micrometer values. The most stable
(a, b) combinations allowed by this resolution constraint were
selected.
3.2. Fabrication. The pillar dimensions and spacings obtained
as vectors a(x) and b(x) defined the gradients. Layouts of these
gradients were drafted using L-Edit. Transparency photomasks
were printed at CAD/Art Services Inc. (Figure 7a). We used
We first present proof-of-principle results to validate the basic
hypothesis. We then present other salient observations. We create
plots of theoretical predictions using our model, compare them
with experimental results, and explain the trends.
5.1. Gradients Guide Droplets. Vibration provided the force
required to overcome pinning and allowed the bases of the droplets
to “explore” their vicinity and sense the gradients. Droplets then
moved down the gradients. Alternatively, we can summarize the
phenomenon as texture-controlled gradients rectifying the random
motion of droplets caused by vibration. To ascertain that the
gradient and not the stage tilt or vibration bias directed the droplet,
we designed a control experiment.
For the control experiment, two similarly sized droplets were
placed on oppositely oriented gradients on the same die. When
vibrated, these Fakir droplets moved as expected, in opposite
directions down their respective gradients (Figure 9). Thus, the
proof-of-principle was established: roughness-controlled gradients can direct Fakir droplets.
The frame that showed the first observable displacement of
the stage marked the beginning of the experiment and was assigned
as t ) 0. The stage height along with the droplet position and
its height were determined in each frame using an image analysis
program that we developed in Matlab. For one such run, Figure
10a shows the position of a 5 µL droplet on the gradient every
2 ms. It illustrates different phases of the journey on the continuous
gradient: acceleration, uniform velocity, deceleration, and an
eventual halt. Figure 10b shows the average over eight such runs
of a 5 µL droplet. Two observations were made: that the runs
were repeatable and that the droplets did not traverse the entire
length of the gradient.
5.2. Droplets Stop When the “Difference in Slopes” Reaches
a Critical Value. On each gradient surface, we carried out several
runs by varying the volume of the droplets but keeping the
frequency and amplitude of vibration constant. We noticed that
droplets of different volumes came to a halt at different location
on the same gradient. To analyze this behavior, we begin by
identifying appropriate variables and then create predicted plots
using the model and finally plot the observations to make a
comparison.
6166 Langmuir, Vol. 22, No. 14, 2006
Shastry et al.
Figure 7. Fabrication process for the test surface. (a) Design of the surface. The (a, b) values for the designed gradients were laid out using
L-Edit and printed to create transparency masks. (b) Fabrication of the surface: Photolithography with an AZ4620 resist was followed by
DRIE using the standard Bosch process. Etch depths ranged from 70 to 80 µm. The wafer was cleaned, and a hydrophobic coating of Teflon
AF 1600 was applied. (c) SEM micrograph of pillars etched in silicon. Each die consisted of two parallel, oppositely oriented, 4-mm-wide
gradients 1 mm apart. The gradient tracks were surrounded by a forbidden region with φ < φmin to prevent droplets from leaving the gradient
track.
Figure 8. (a) Droplet surface vibrated by a speaker powered by a function generator set at 20 Hz with a 50 mV amplitude. A Gemini
high-intensity light source illuminates the droplet. A DRS camera records the movies at 500 fps. (b) Variation of the footprint of a 5 µL
droplet for the square wave excitation of the speaker shown in part a.
Figure 9. Droplets on oppositely oriented gradients on the same
die, when vibrated, traveling in opposite directions down their
respective gradients. The droplets were similar in volume (left, 7.67
µL; right, 6.35 µL).
For a given droplet size, the driving force that it experiences
is determined by the slope and the random force due to vibration.
Hysteresis, which is dependent on the surface structure, determines
the impeding force. We employ the model developed in eq 16
to plot the slope required for a 5 µL droplet to begin to move
on the continuous gradient shown in Figure 5. The actual slope
is also plotted; it is evident that the droplet cannot move without
the mitigating help of vibration. For the purpose of this discussion,
we call the difference between the slope required for incipient
movement and the actual slope at a given location as the
“difference of slopes” or simply the “gap”. The gap for a droplet
at a location on a gradient is a measure of the additional force
required to make it move. We use our model to create Figure
11a, which is a plot of gap versus location on the gradient for
different droplet volumes. As the gap increases, more force will
be required to make the droplet move. Alternately, if we apply
a constant force to the droplet, then the droplet will stop when
the gap increases to an “upper-limit” value. It can be seen that
the curves corresponding to different volumes reach the specific
upper limit of 0.68 at different locations. We argue that this
“upper-limit” value of 0.68 is a measure of the force applied to
the droplets due to vibration. We hypothesize that vibration
supplies droplets with sufficient force to overcome a critical gap
value of less than 0.68; therefore, the points of intersection in
Figure 11a denote the locations on the gradient where the droplets
of various volumes come to a halt. Strictly, the force balance is
Directing Droplets Using Microstructured Surfaces
Langmuir, Vol. 22, No. 14, 2006 6167
Figure 10. (a) Position of the center of a 4.96 µL droplet plotted as a function of time. Black diamonds mark the movie frames shown at
the right as the droplet moves down the 6.24-mm-long gradient. A black ruler above and below the images indicates the length and location
of the gradient in the images. (b) Average position of various droplets on the gradient plotted as a function of time. We plot data for the
part of the gradient that all droplets have in common (i.e., after the first 1.45 mm of the gradient).
Figure 11. (a) Difference between the slope required for incipient motion and the actual value of the gradient (gap) as a function of φ(x)
for the continuous gradient. A larger gap implies that a larger additional force is required to move the droplet. Smaller droplets had larger
gaps at each location; they reached the critical value for the gap (∆dφ/dx ) 0.68) earlier in the gradient than did larger droplets. (b) Solid
line showing the predicted stop location (where the gap first increased to the critical value of 0.68) plotted as a function of droplet volume.
Experimentally observed stop locations of the droplets, measured at the center of the droplet, are plotted as red points. The gray regions indicate
the effective end of the gradient (i.e., if the center of the droplet were positioned in these regions, then from eq 17 it is evident that part
of the droplet base could cross the end of the gradient).
valid at the point where the droplet just begins to decelerate. We
make simplifying assumptions that the force balance is valid at
the stopping location, neglecting the drag force and the distance
over which the droplet decelerates to a halt.
To test our hypothesis, we plot a curve using these predicted
“stop” locations of the various sized droplets as a function of
their volume, as shown in Figure 11b. We plot the experimentally
obtained stop positions for droplets of several volumes. As evident
from Figure 11b, the good fit validates the capability of our
analytical model to predict trends.
6. Conclusions
This work provides design rules for exploiting physical texture,
specifically the solid-liquid contact area fraction φ, to create
surface-energy gradients that guide droplets along desired
trajectories. We have developed an analytical model based on
the current theory of wetting of rough surfaces. This model
accounts for contact angle hysteresis and predicts the slope
required for spontaneous movement of the droplet. We have
demonstrated proof-of-principle results along with some trends
and showed that the observed trends can be predicted from the
model.
While this work introduces hard-coded gradients where droplets
are propelled by vibration, it also lays the foundation for
completely programmable gradients. Electrically controlled
gradients steep enough to overcome hysteresis would move with
the droplet to control it. Electrowetting could be the choice of
technology to dynamically control the solid-liquid area of Fakir
droplets,33 eventually leading to completely programmable
CMOS-compatible microfluidic platforms.
Acknowledgment. This work is funded by NIH Center of
Excellence in Genomic Science and Technology grant 1-P50HG002360-01. A.S. thanks Sidhartha Goyal for useful technical
discussions. M.J.C. thanks the UWEB REU program at the
University of Washington, which made this association possible.
K.F.B. acknowledges a fellowship from the Japan Society for
the Promotion of Science (JSPS). He thanks M. Esashi (Tohoku
University), H. Fujita (University of Tokyo), and O. Tabata
(University of Kyoto) for their generous hospitality during his
stay at their laboratories.
LA0601657
(33) Krupenkin, T. N.; Taylor, J. A.; Schneider, T. M.; Yang, S. Langmuir
2004, 20, 3824-3827.